CN111752149A - Design method of digital repetitive controller for designated time attraction of servo motor - Google Patents

Design method of digital repetitive controller for designated time attraction of servo motor Download PDF

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CN111752149A
CN111752149A CN202010510384.7A CN202010510384A CN111752149A CN 111752149 A CN111752149 A CN 111752149A CN 202010510384 A CN202010510384 A CN 202010510384A CN 111752149 A CN111752149 A CN 111752149A
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attraction
time
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CN111752149B (en
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孙明轩
朱鹏升
薛良曾
邹胜祥
李威
李杏
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Zhejiang University of Technology ZJUT
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Abstract

A servo motor appointed time attraction digital repetitive controller design method comprises the steps that a given module generates a periodic reference signal and a periodic feedback link is constructed; equivalent disturbance compensation is introduced into an attraction law of a specified time, and an interference observer is used for estimating equivalent disturbance; constructing an ideal error dynamic state (an attraction law with interference suppression), designing a controller according to the ideal error dynamic state, and taking a signal obtained by current calculation as the input of a servo system; the specific controller parameter setting is carried out according to the convergence performance index of the representation system, and a calculation formula of an attraction domain, a monotone decreasing region, an absolute attraction layer, a steady-state error band and a maximum convergence step number in the process of representing the convergence of the tracking error is provided. The designated time attraction repetitive controller provided by the invention can realize complete suppression of periodic components of interference signals and suppress the influence of non-periodic components.

Description

Design method of digital repetitive controller for designated time attraction of servo motor
Technical Field
The invention relates to a design method of a repetitive controller attracted by appointed time, which can be used for repetitive tracking control of a servo motor and is also suitable for other industrial processes operating periodically.
Background
There are a number of control systems in industrial applications that track periodic reference signals, such as servo motors that perform reciprocating operations, hard disk drive servo systems, sinusoidal alternating power electronic systems, and the like. The existing repetitive control technology mainly focuses on a frequency domain analysis and design method based on an internal model principle, and a dynamic model of an external action signal is implanted into a controller to form a high-precision feedback control system. The internal model principle states that in any feedback control system that can cancel external disturbances or track a reference input signal, the feedback loop must contain a dynamic model that is identical to the external input signal. In order to completely eliminate the influence of external disturbance on the performance of a control system and enable the system to realize the tracking of the unsteady state error of any form of reference input signals. For continuous time systems, the repetitive controller constructs a periodic signal internal model
Figure BDA0002528166390000011
Wherein T isCIs the period of the reference signal, which is a time delay with period
Figure BDA0002528166390000012
The positive feedback link of (1). Regardless of the specific form of the reference signal, as long as the initial segment signal is given, the internal mode output can accumulate the input signal cycle by cycle, and repeatedly output the signal with the same cycle as the previous cycle to form the reference signal. The continuous repetitive controller frequency domain design employs this continuous internal model. In practice, the motor computer control technology under the periodically symmetrical reference signals is adopted, and most systems are realized in a discrete mode. The digital repetitive controller in the invention is designed to realize the dispersion of signals by sampling. Taking the sampling period as TsMaking the period of reference signal be integral multiple of sampling period, and recording the number of sampling points in each period as N, i.e. TC＝NTs. Thus, the discrete periodic signal is internally modeled as
Figure BDA0002528166390000013
The computational complexity of the discrete internal model is mainly determined by the sampling period TsThe amount of memory required to implement the discrete period internal model is proportional to N. If T issThe control precision of the system is reduced when the acquisition is too large; if the extraction is too small, the order of the internal mold will increase.
The common attraction law can reflect the error attenuation characteristic, the attraction law method is a control system design method which directly utilizes the tracking error and enables the tracking error to be converged according to a preset attraction mode, and the controller design is more direct and concise. From published documents, the existing limited-time convergence laws of attraction are not abundant, and the discovery of novel laws of attraction is important for gradually enriching the design method with continuously improved convergence performance. The attraction law method directly adopts the tracking error signal without defining a switching function, and the design of the controller is more direct and simpler. The attraction law reflects the expected dynamic characteristics of the system error when disturbance is not considered; in the presence of disturbances, a controller directly based on the attraction law cannot be implemented. The interference suppression measures can be 'embedded' into the attraction law, and ideal error dynamics with the disturbance suppression effect are constructed. And designing the discrete time controller according to the constructed ideal error dynamic equation, wherein the dynamic process of the closed-loop system is determined by the ideal error dynamic, and the expected tracking performance represented by the ideal error dynamic is realized. The attraction law method is different from an approach law method of discrete sliding mode control. The main differences between the two are as follows: the attraction law method replaces the tracking error with a switching function and replaces the original point with a switching surface; the approach law method requires a finite time to reach the switching surface, while the attraction law method requires a finite time to reach the origin; the closed-loop system designed by the attraction law method still has robustness performance related to parameter drift and external interference, only the sliding mode control emphasizes invariance of sliding mode motion, and the attraction law method pursues invariance of system steady state. When the repetitive controller is designed by an attraction law method, indexes describing transient and steady-state behaviors of the tracking error can be dynamically given by an ideal error, and the indexes comprise the following five indexes: an attraction domain, a steady state error band, an absolute attraction layer, a monotonically decreasing region, and a maximum number of convergence steps required for the tracking error to first enter the steady state error band. In fact, the specific values of the five indexes depend on the controller parameters, the controller parameters are different, and the values of the five indexes are also different. Once the ideal error dynamic form is given, specific expressions of the four indexes can be given in advance for controller parameter tuning. In the currently published methods of attraction law, the main criteria depend on the boundary of the equivalent interference signal. The boundary that effectively inhibits interference and reduces equivalent interference signals is a difficult problem to be solved urgently by an attraction law method.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a method for designing a repetitive controller of designated time suction, which aims to enable a closed-loop system to have preset expected error tracking performance and effectively suppress buffeting, provides a suction law capable of converging designated time, and designs a servo motor repetitive controller according to an ideal error dynamic equation constructed by the suction law. When the periodic interference component is completely inhibited and the non-periodic component in the disturbance is considered, an equivalent disturbance observer is introduced into a closed-loop system and is used for compensating the non-periodic interference so as to improve the control performance and enable a servo system to realize high-speed and high-precision tracking.
The invention adopts the technical scheme for solving the technical problems that:
a design method of a repetitive controller for designated time attraction of a servo motor comprises the following steps:
1) setting a reference signal whose periodicity satisfies
rk＝rk-N(1)
Wherein r iskAnd rk-NRespectively representing reference signals at the k moment and the k-N moment, wherein N is the period of the reference signals;
2) the tracking error signal is defined and,
ek＝rk-yk(2)
wherein e iskIndicating the tracking error at time k, ykRepresenting the system output at time k;
3) given law of continuous attraction
Figure BDA0002528166390000021
Wherein, the adjustable parameter rho is more than 0, alpha is more than 0 and less than 1, mu is more than 0, e represents continuous tracking error signal, formula (3) is the designated time suction law, and the convergence time is
Figure BDA0002528166390000022
Wherein e is0The initial value of the tracking error is expressed, and it can be found from the expression (4) that the convergence time of the attraction law is definite, that is, the fixed time T is providedmax(mu) satisfy
Figure BDA0002528166390000023
Discretizing the formula (3) to construct a discrete attraction law
Figure BDA0002528166390000024
Wherein the adjustable parameters rho is more than 0, 0 is more than α and less than 1, mu is more than 0, ek+1Represents the tracking error at time k + 1;
4) structural equivalent disturbance
dk＝wk-wk-N(7)
Wherein d iskRepresenting the equivalent disturbance signal at time k, wkAnd wk-NRespectively representing the system interference signals at the k moment and the k-N moment, and the relation of equivalent disturbance and tracking error is
ek+1＝rk+1-yk+1＝rk+1-yk+1-N+A′(q-1)(yk-yk-N)-q-d+1B(q-1)(uk-uk-N)-dk+1(8)
In the formula,
A′(q-1)＝a1+a2q-1+…+anq-n+1＝q(A(q-1)-1)
A(q-1)＝1+a1q-1+…+anq-n
B(q-1)＝b0+b1q-1+…+bmq-m
dynamic characteristic model satisfying following servo motor
A(q-1)yk＝q-dB(q-1)uk+wk(9)
Wherein u iskAnd ykRespectively representing input and output signals at time k, dk+1Representing the equivalent disturbance at the moment k + 1; a (q)-1) And B (q)-1) Is q-1Polynomial of (a), q-1Is a one-step delay operator, n represents A (q)-1) M represents B (q)-1) Order of (a)1,…,an,b0,…,bmRepresents a system parameter and b0Not equal to 0, n is more than or equal to m; d is an integer, and d is an integer more than or equal to 1;
5) equivalent disturbance estimation
Designing an observer to estimate the equivalent disturbance and compensating the equivalent disturbance by an observed value, wherein two observed variables of the observer are
Figure BDA0002528166390000031
And
Figure BDA0002528166390000032
are each ekAnd dkTo this end, the following observer is constructed
Figure BDA0002528166390000033
Wherein, β1Indicating observer gain coefficient with respect to error, β2Representing the observer gain coefficient with respect to the equivalent disturbance,
Figure BDA0002528166390000034
representing the error signal ek+1Is estimated by the estimation of (a) a,
Figure BDA0002528166390000035
representing an equivalent disturbance dk+1Is estimated by the estimation of (a) a,
Figure BDA0002528166390000036
for estimation error of tracking error, when k is large enough, for assurance
Figure BDA0002528166390000037
And
Figure BDA0002528166390000038
respectively converge on ekAnd dkNeed to configure parameters β1，β2Make the matrix
Figure BDA0002528166390000039
All eigenvalues of (a) are within the unit circle;
6) constructing laws of attraction with interference suppression
Figure BDA00025281663900000310
7) Repetitive controller design
According to the law of attraction with interference suppression,
Figure BDA00025281663900000311
thus, the repetitive controller expression obtained is
Figure BDA00025281663900000312
Obtaining the control signal u of the servo object at the time k by the above formulakWherein the reference signal rkAnd rk+1Generated by a given module; using measured servo system output signal ykCalculating a tracking error ek(ii) a Signal uk-N、yk+1-N、yk-NGiven by the memory module.
Further, heavy in weightAfter the complex controller design is completed, the equivalent disturbance margin delta is defined, i.e.
Figure BDA0002528166390000041
The specific controller parameter setting can be carried out according to indexes representing the convergence performance of the system; in order to represent the convergence performance of the tracking error, the performance indexes introduced by the method comprise a monotone decreasing area, an absolute attraction layer, a steady-state error band and a maximum convergence step number; further, the concept of an attraction domain is introduced for describing the convergence range of the attraction law, and when the monotone subtraction region, the absolute attraction layer and the steady-state error band are located in the attraction domain, corresponding boundary values exist, which are defined as follows:
attraction domain boundary ΔAB: namely the compression condition satisfied by the attraction law;
monotonous decreasing region deltaMD: when e iskGreater than this boundary, ekThe same number is decreased, namely the following conditions are met:
Figure BDA0002528166390000042
absolute attraction layer ΔAA: absolute value of system tracking errorkIf | is greater than this boundary, its | ekI, monotonically decreases, i.e. the following condition is satisfied:
Figure BDA0002528166390000043
steady state error band ΔSS: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:
Figure BDA0002528166390000044
maximum number of convergence steps
Figure BDA0002528166390000045
The tracking error passes through at most
Figure BDA0002528166390000046
Entering a steady state error band;
when the equivalent disturbance compensation error is satisfied
Figure BDA0002528166390000047
The expression of each index is as follows:
attraction domain boundary ΔAB
Figure BDA0002528166390000048
Monotonous decreasing region deltaMD
ΔMD＝max{ΔMD1,ΔMD2} (18)
Figure BDA0002528166390000049
Wherein, DeltaMD1And ΔMD2Are all positive and real, and are determined by equation (19);
absolute attraction layer ΔAA
ΔAA＝max{ΔAA1,ΔAA2} (20)
Figure BDA00025281663900000410
Wherein, DeltaAA1And ΔAA2Are all positive real numbers and are determined by equation (21);
steady state error band ΔSS
ΔSS＝max{ΔSS1，ΔSS2} (22)
Figure BDA0002528166390000051
Wherein, DeltaSS1And ΔSS2Are all positive real numbers and are determined by equation (23);
maximum number of convergence steps
Figure BDA0002528166390000052
Figure BDA0002528166390000053
Wherein,
Figure RE-GDA0002629228840000054
represents a positive integer not less than ·.
Still further, the adjustable parameters of the controller include ρ, α, and μ; and the parameter setting of the controller is carried out according to the index representing the convergence performance of the system.
When the reference signal satisfies rk＝rk-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is dk＝wk-wk-1(ii) a Wherein r isk-1Representing the reference signal at time k-1, wk-1Representing the interference signal at the k-1 moment; the feedback controller with equivalent disturbance compensation is
Figure BDA0002528166390000055
The adjustable parameters of the observer include β1And β2When k is sufficiently large, to ensure
Figure BDA0002528166390000056
And
Figure BDA0002528166390000057
respectively converge on ekAnd dkNeed to configure parameters β1，β2Make the matrix
Figure BDA0002528166390000058
Is within the unit circle.
The invention has the technical idea that a discrete repetitive controller of a servo motor is designed according to a designated time attraction law. The design method is visual and simple, is a time domain design method and is different from the currently and generally adopted frequency domain method. The periodic form of a given reference signal is taken into account in the design of the controller, which effectively utilizes the system period tracking characteristics. The time domain design of the controller is easy to combine with the existing interference suppression means, equivalent disturbance observation is added, complete suppression of the periodic component of the interference signal can be realized, the influence of the non-periodic component of the interference signal is suppressed, and the rapid high-precision tracking of the given reference signal is realized.
The invention has the main effects that: specified convergence performance, effective interference suppression and high control accuracy.
Drawings
Fig. 1 is a block diagram of a servo motor control apparatus.
FIG. 2 is a block diagram of an equivalent disturbance observer.
Fig. 3 is a block diagram of a repetitive controller for a timing attraction law.
Fig. 4 shows the parameters ρ 0.3, α 0.9, μ 10, e0The time attraction law convergence time is continuously specified at 1.
Fig. 5 shows Δ when the repetitive controller parameter is ρ 0.05, α 0.3, and μ 1MDTrend with Δ.
Fig. 6 shows Δ when the repetitive controller parameter is ρ 0.05, α 0.3, and μ 1AATrend with Δ.
Fig. 7 shows Δ when the repetitive controller parameter is ρ 0.05, α 0.3, and μ 1SSTrend with Δ.
FIG. 8 is a graph of the time when a disturbance w occursk＝sin(2πfkTs) +0.15sgn (sin (2k pi/150)), and the boundary value Δ when the repetitive controller parameter is ρ 0.05, α 0.2, μ 1, and Δ 0.3AB，ΔMD，ΔAAAnd deltaSS。
FIG. 9 is a graph of the time when a disturbance w occursk＝sin(2πfkTs) +0.15sgn (sin (2k pi/150)), and the boundary value Δ when the repetitive controller parameter is ρ 0.05, α 0.5, μ 1, and Δ 0.3AB，ΔMD，ΔAAAnd deltaSS。
Fig. 10 shows the parameters ρ 0.1, α 0.2, e0When μ ═ 1, μ ═ 5, and μ ═ 10 continuous suctionLaw convergence time and corresponding Tmax(1)、Tmax(5) And Tmax(10)。
Fig. 11 shows the parameters ρ 0.1, α 0.2, e0When the value is 1, the variation of the difference Δ T between the continuous suction law fixed time and the convergence time with μ tends to vary.
Fig. 12 to 15 show experimental results of the permanent magnet synchronous motor control device when the repetitive controller parameter ρ is 0.03, α is 0.1, and μ is 3, wherein,
FIG. 12 is a reference signal and system output under the influence of a repetitive controller based on the prescribed time law of attraction;
FIG. 13 is a control input under the influence of a repetitive controller based on a prescribed temporal attraction law;
FIG. 14 is a tracking error under the influence of a repetitive controller based on the attraction law for a given time;
fig. 15 is a tracking error distribution histogram under the action of the repetitive controller based on the specified time attraction law.
Fig. 16 to 19 show equivalent disturbance observer parameters β when the repetitive controller parameters ρ is 0.03, α is 0.1, and μ is 31＝0.2， β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 16 is a reference signal and system output under the influence of a repetitive controller based on a specified time law of attraction and equivalent disturbance compensation;
FIG. 17 is a control input under the influence of a repetitive controller based on the specified time law of attraction and equivalent disturbance compensation;
FIG. 18 is a graph of tracking error under repetitive controller action based on the specified time law of attraction and equivalent perturbation compensation;
FIG. 19 is a histogram of the tracking error distribution under the influence of a repetitive controller based on the specified time law of attraction and equivalent disturbance compensation.
Fig. 20 to 23 are experimental results of the permanent magnet synchronous motor control device when the feedback controller parameter is ρ ═ 0.03, α ═ 0.1, and μ ═ 3, where:
FIG. 20 is a reference signal and system output under the influence of a feedback controller based on the attraction law for a given time;
FIG. 21 is a control input based on a feedback controller for a specified time attraction law;
FIG. 22 is a graph of tracking error under the influence of a feedback controller based on the attraction law for a given time;
fig. 23 is a tracking error distribution histogram under the action of a feedback controller based on a specified time attraction law.
Fig. 24-27 show equivalent disturbance observer parameters β when the feedback controller parameters ρ is 0.03, α is 0.1, and μ is 31＝0.2， β20.5, experimental results of the permanent magnet synchronous motor control device, wherein:
FIG. 24 is a reference signal and system output under the influence of a feedback controller based on the assigned time law of attraction and equivalent disturbance compensation;
FIG. 25 is a control input under the influence of a feedback controller based on the specified time law of attraction and equivalent disturbance compensation;
FIG. 26 is a graph of tracking error under the influence of a feedback controller based on the attraction law for a given time and equivalent disturbance compensation;
FIG. 27 is a histogram of the tracking error distribution under the influence of a feedback controller based on the specified time law of attraction and equivalent disturbance compensation.
Fig. 28 to 29 show that ρ is 0.03, α is 0.1, μ is 3, and the equivalent disturbance observer parameter is β1＝0.2，β2And when the tracking error is 0.5, the tracking error is caused by the repetitive controller based on the designated time attraction law and equivalent disturbance compensation.
Fig. 30-31 show the controller parameters ρ 0.03, α 0.1, μ 3, and the equivalent disturbance observer parameters β1＝0.2，β2And when the tracking error is equal to 0.5, the tracking error is generated under the action of a repetitive controller based on the designated time attraction law and the equivalent disturbance compensation and a feedback controller based on the designated time attraction law and the equivalent disturbance compensation.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings.
Referring to fig. 1 to 31, a method for designing a servo motor timing attraction law repetitive controller, wherein fig. 1 is a block diagram of a servo motor control device; FIG. 2 is a block diagram of an equivalent disturbance observer; fig. 3 is a schematic diagram of a repetitive controller structure of the timing attraction law.
A design method of a repetitive controller applying equivalent disturbance compensation to a servo motor specified time attraction law comprises the following steps:
step 1, setting a reference signal, wherein the periodicity of the reference signal meets (1);
step 2, defining a tracking error (2);
step 3, giving a continuous attraction law (3);
step 4, discretizing the continuous attraction law to construct a discrete attraction law (6);
step 5, constructing an equivalent disturbance (7);
designing an observer for equivalent disturbance estimation;
step 7, constructing an attraction law (11) with interference suppression effect;
and 8, designing a repetitive controller (13) with equivalent disturbance compensation.
Further, the repetitive controller design method is explained as follows:
introduction of d into attraction law of appointed timek+1Reflecting the suppression measures for a given periodic disturbance signal, introducing
Figure BDA0002528166390000071
And giving an estimated value of the equivalent disturbance for compensating the equivalent disturbance.
The estimation of the equivalent disturbance uses a one-step delay estimate or median estimate, the one-step delay estimate being
Figure BDA0002528166390000072
A determination method when the median value is estimated to be known, and d is respectively set as the upper and lower bounds of the equivalent disturbanceuAnd dlMemory for recording
Figure BDA0002528166390000073
Δ＝(du-dl) A/2, then
Figure BDA0002528166390000074
Is convenient to use
Figure BDA0002528166390000075
Equation (13), reference Signal rkAnd rk+1Generated by a given module; using measured servo system output signal ykThe tracking error e can be calculatedk(ii) a Signal uk-1、uk-1-N、uk-N、yk-1、yk-1-N、yk-NCan be read from the memory.
When the reference signal satisfies rk＝rk-1The discrete repetitive controller is also suitable for constant regulation, where the equivalent disturbance is dk＝wk-wk-1Wherein r isk-1Reference signal representing the time instant k-1, wk-1Representing the interference signal at time k-1, a feedback controller with equivalent disturbance compensation is
Figure BDA0002528166390000076
The above-mentioned repetitive controller can also give the design result of a higher-order system in the same way for a second-order system.
Further, after the controller is constructed, the controller also needs to be based on the attraction domain deltaABMonotonous decreasing region deltaMDAbsolute attraction layer DeltaAASteady state error band deltaSSAnd maximum number of convergence steps k*And setting the controller parameters to achieve the optimal control effect.
In this embodiment, a servo system of a permanent magnet synchronous motor is taken as an example, and a position reference signal of a korean LS ac servo motor APM-SB01AGN is taken as a control object. The ELMO AC servo driver and the TMS320-F2812DSP development board are used as controllers, three-loop control is adopted, a current loop and speed loop controller is provided by the ELMO driver, and a position loop is provided by the DSP development board. The upper computer is used for process monitoring and data storage.
The position loop controller is designed, and a mathematical model of a servo object except the position loop is required to be established, wherein the mathematical model comprises a current loop, a speed loop, a power driver, an alternating current permanent magnet synchronous servo motor body and a detection device. The mathematical model of the servo object is obtained by the system identification least square algorithm
yk+1-1.8949yk+0.8949yk-1＝1.7908uk-0.5704uk-1+wk+1(27)
Wherein, ykAnd ukPosition output and velocity set signal (control input), w, respectively, for a position servo systemk+1Representing the interfering signal.
The present embodiment will illustrate the effectiveness of the repetitive controller design method provided by the present invention through numerical verification and experimental results, respectively.
With the sinusoidal signal as the position reference signal, the repetitive controller takes the form of a controller given by equation (13), a specific expression of which can be written as
Figure BDA0002528166390000081
First, the accuracy of the convergence time of the continuous attraction law is shown by the numerical results, where the initial error e is set by setting the parameters ρ 0.3, α 0.9, μ 100The convergence time t is calculated according to equation (4) as 0.285s at 1, and the numerical simulation results are shown in fig. 4, which shows the convergence of the error variable at t as 0.285 s.
Under the action of the repetitive controller (28), the same controller parameters rho, α and mu are selected, different delta is selected, and the three boundary layers of the servo system are changed correspondinglyMDAbsolute attraction layer DeltaAAAnd steady state error band ΔSSGiven the controller parameters, rho is 0.05, α is 0.3, and mu is 1, the simulation results are shown in figures 5-7, and under the condition of a given system model and a reference signal, the graph results show the variation trend of each boundary value along with the equivalent disturbance boundary.
Given a position reference signal of rk＝20sin(2πfkTs) In degrees (deg), frequency f 1Hz, and sampling time TsThe number of times of one periodic sampling N is 1000. During simulation, the selected disturbance amount wkIs composed of periodic disturbance and non-periodic random disturbance in the form of
wk＝sin(2πfkTs)+0.15sgn(sin(2kπ/150)) (29)
Under the action of the repetitive controller (28), different controller parameters rho, α and mu are selected, boundary layers of servo system convergence processes are different, and in order to verify the attraction domain delta given by the patent of the inventionABMonotonous decreasing region deltaMDAbsolute attraction layer DeltaAAAnd steady state error band ΔSSThe boundary layer expression of (2) is subjected to numerical simulation.
1) When the controller parameter ρ is 0.05, α is 0.2, μ is 1, and Δ is 0.3, according to the calculation formula of the attraction domain boundary and the other three boundary values,
ΔMD＝2.9489，ΔSS＝ΔAA＝0.2948
the range of the attraction domain is 0.0021 < deltaAB< 337.6846. From the above data, it is understood that three boundary values exist within the range of the attraction domain, i.e., three boundaries exist.
2) When the controller parameter ρ is 0.05, α is 0.5, μ is 1, and Δ is 0.3, according to the calculation formula of the attraction domain boundary and the other three boundary values,
ΔSS＝ΔAA＝ΔMD＝0.7739
the range of the attraction domain is 1 × 10-4＜ΔAB< 11.9821. From the above data, it is understood that three boundary values exist within the range of the attraction domain, i.e., three boundaries exist.
The simulation results are shown in fig. 8-9. The numerical results verify the attraction domain delta of the tracking error of the system under the action of the repetitive controller given by the patent under the condition of a given system model, a reference signal and a disturbance signalABMonotonous decreasing region deltaMDAbsolute absorption layer DeltaAAAnd steady state error band ΔSSExpression formula。
This example illustrates the necessity of introducing linear terms in the designated time attraction law of the present invention by numerical validation.
According to equations (4) and (5), the parameters ρ 0.1, α 0.2, e are given0When the value is 1, the convergence time of continuous suction law of mu-1, mu-5 and mu-10 and the corresponding fixed time Tmax(1)、Tmax(5) And Tmax(10) The numerical simulation results are shown in fig. 10, which shows that as μ increases, the convergence time decreases, the settling time decreases, and the actual convergence time and settling time are closer together.
According to equations (4) and (5), the parameters ρ 0.1, α 0.2, e are given0When the value is 1, μ is an acceleration term coefficient of a predetermined time attraction law, and Δ T represents a difference between the attraction law fixed time and the convergence time, that is, Δ T is Tmax(μ)-tsAt the parameters ρ, α, e0In a certain case, as the acceleration term μ increases, the convergence time gradually approaches the fixed time, and Δ T becomes smaller, and fig. 11 shows a variation trend graph of Δ T with μ.
The block diagram of the servo motor control device used in the experiment is shown in fig. 1, and is used for verifying the tracking performance of the discrete controller of the designated time attraction law. Given a reference trajectory r (k) ═ a (sin (2 pi (k-200)/N) +1) where the amplitude a equals 135, the sampling period Ts2.5ms, k is the number of samples, and N is 800.
The repetitive controller employed is as follows
Figure BDA0002528166390000091
The repetitive controller adopting disturbance compensation based on the equivalent disturbance observer is as follows
Figure BDA0002528166390000092
The adopted feedback controller adopts the following steps
Figure BDA0002528166390000093
The feedback controller adopting disturbance compensation based on the equivalent disturbance observer is as follows
Figure BDA0002528166390000101
The experimental results using the above controller are as follows:
1) the repetitive controller (30) is adopted, the controller parameters are rho is 0.03, alpha is 0.1, and mu is 3, and the experimental results are shown in fig. 12-15.
2) Adopting a repetitive controller (31), wherein the controller parameters are rho 0.03, α 0.1 and mu 3, and the equivalent disturbance observer parameter β1＝0.2，β2The results are shown in fig. 16-19, at 0.5.
3) A feedback controller (32) is adopted, controller parameters are rho-0.03, alpha-0.1 and mu-3, and the experimental results are shown in fig. 20-23.
4) Adopting a feedback controller (33), wherein the controller parameters are rho 0.03, α 0.1 and mu 3, and equivalent disturbance observer parameters β1＝0.2，β2The results are shown in fig. 24-27, 0.5.
From the experimental results it can be seen that:
the repetitive controller (30) and the repetitive controller (31) are used for experimental comparison, the repetitive control can completely inhibit periodic disturbance, but the tracking performance deviation of the first period is avoided, and the disturbance observer improves the tracking performance of the repetitive controller in the first period and reduces the influence of non-periodic disturbance on the tracking performance of the system;
the feedback controller (32) and the feedback controller (33) are used for experimental comparison, the feedback control cannot realize complete suppression of periodic disturbance, and the disturbance observer greatly reduces the influence of the periodic disturbance and non-periodic disturbance on the tracking performance of the system;
the repetitive controller (30) and the feedback controller (32) are used for experimental comparison, and the tracking error of the latter has obvious periodicity obviously from the experimental result.
Further, the comparison of tracking performance under the condition that the equivalent disturbance observer is adopted or not by the repetitive controller, and the equivalent disturbance observer is respectively adopted by the repetitive controller and the feedback controller is shown in fig. 28-31.
The experimental result shows that the repetitive control method can completely inhibit periodic disturbance, and the periodic tracking performance of the servo motor is obviously improved. In addition, equivalent disturbance is introduced, an equivalent disturbance observer is used for estimating the system, a compensation effect is provided in the controller, the influence of unknown disturbance on the tracking performance can be effectively inhibited, and the tracking performance of the system is further improved.

Claims (5)

1. A design method of a digital repetitive controller for designated time attraction of a servo motor is characterized by comprising the following steps:
1) setting a reference signal whose periodicity satisfies
rk＝rk-N(1)
Wherein r iskAnd rk-NRespectively representing reference signals at the k moment and the k-N moment, wherein N is the period of the reference signals;
2) the tracking error signal is defined and,
ek＝rk-yk(2)
wherein e iskIndicating the tracking error at time k, ykRepresenting the system output at time k;
3) given law of continuous attraction
Figure FDA0002528166380000011
Wherein, the adjustable parameter rho is more than 0, alpha is more than 0 and less than 1, mu is more than 0, e represents continuous tracking error signal, formula (3) is the designated time suction law, and the convergence time is
Figure FDA0002528166380000012
Wherein e is0The initial value of the tracking error is expressed, and it can be found from the equation (4) that the convergence time of the attraction law is definite, i.e., has a fixed time Tmax(mu) satisfy
Figure FDA0002528166380000013
Discretizing the formula (3) to construct a discrete attraction law
Figure FDA0002528166380000014
Wherein the adjustable parameters rho is more than 0, 0 is more than α and less than 1, mu is more than 0, ek+1Represents the tracking error at time k + 1;
4) structural equivalent disturbance
dk＝wk-wk-N(7)
Wherein d iskRepresenting the equivalent disturbance signal at time k, wkAnd wk-NRespectively representing the system interference signals at the k moment and the k-N moment, and the relation of equivalent disturbance and tracking error is
ek+1＝rk+1-yk+1＝rk+1-yk+1-N+A′(q-1)(yk-yk-N)-q-d+1B(q-1)(uk-uk-N)-dk+1(8)
In the formula,
A′(q-1)＝a1+a2q-1+…+anq-n+1＝q(A(q-1)-1)
A(q-1)＝1+a1q-1+…+anq-n
B(q-1)＝b0+b1q-1+…+bmq-m
dynamic characteristic model satisfying following servo motor
A(q-1)yk＝q-dB(q-1)uk+wk(9)
Wherein u iskAnd ykRespectively representing input and output signals at time k, dk+1Representing the equivalent disturbance at the moment k + 1; a (q)-1) And B (q)-1) Is q-1Polynomial of (a), q-1Is a one-step delay operator, n represents A (q)-1) M represents B (q)-1) Order of (a)1,…,an,b0,...,bmRepresents a system parameter and b0Not equal to 0, n is more than or equal to m; d represents a delay, and d.gtoreq.1;
5) equivalent disturbance estimation
Designing an observer to estimate the equivalent disturbance and compensating the equivalent disturbance by an observed value, wherein two observed variables of the observer are
Figure FDA0002528166380000021
And
Figure FDA0002528166380000022
are each ekAnd dkTo this end, the following observer is constructed
Figure FDA0002528166380000023
Wherein, β1Indicating observer gain coefficient with respect to error, β2Representing the observer gain coefficient with respect to the equivalent disturbance,
Figure FDA0002528166380000024
representing the error signal ek+1Is estimated by the estimation of (a) a,
Figure FDA0002528166380000025
representing an equivalent disturbance dk+1Is estimated by the estimation of (a) a,
Figure FDA0002528166380000026
for estimation error of tracking error, when k is large enough, for assurance
Figure FDA0002528166380000027
And
Figure FDA0002528166380000028
respectively converge on ekAnd dkNeed to configure parameters β1，β2Make the matrix
Figure FDA0002528166380000029
All eigenvalues of (a) are within the unit circle;
6) construction of attraction law with interference suppression
Figure FDA00025281663800000210
7) Repetitive controller design
According to the law of attraction with interference suppression,
Figure FDA00025281663800000211
thus, the repetitive controller expression obtained is
Figure FDA00025281663800000212
Obtaining the control signal u of the servo object at the time k by the above formulakWherein the reference signal rkAnd rk+1Generated by a given module; using measured servo system output signal ykCalculating a tracking error ek(ii) a Signal uk-N、yk+1-N、yk-NGiven by the memory module.
2. The design method of the repetitive controller for designated time attraction of the servo motor as claimed in claim 1, characterized in that: after the repetitive controller design is complete, the equivalent disturbance margin Δ is defined, i.e.
Figure RE-FDA00026292288300000213
The specific controller parameter setting is carried out according to the index of the convergence performance of the representation system, and the performance index is introduced for representing the convergence performance of the tracking errorA monotone decreasing area, an absolute attraction layer, a steady-state error band and a maximum convergence step number are arranged; further, the concept of an attraction domain is introduced for describing the convergence range of the attraction law, and when the monotone subtraction region, the absolute attraction layer and the steady-state error band are located in the attraction domain, corresponding boundary values exist, which are defined as follows:
attraction domain boundary ΔAB: namely the compression condition satisfied by the attraction law;
monotonous decreasing region deltaMD: when e iskGreater than this boundary, ekThe same number is decreased, namely the following conditions are met:
Figure RE-FDA0002629228830000031
absolute attraction layer ΔAA: absolute value of system tracking errorkIf | is greater than this boundary, its | ekI, monotonically decreases, i.e., the following condition is satisfied:
Figure RE-FDA0002629228830000032
steady state error band ΔSS: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:
Figure RE-FDA0002629228830000033
maximum number of convergence steps
Figure RE-FDA0002629228830000034
The tracking error passes through at most
Figure RE-FDA0002629228830000035
Entering a steady state error band;
when the equivalent interference compensation error is satisfied
Figure RE-FDA0002629228830000036
The expression of each index is as follows:
attraction domain boundary ΔAB
Figure RE-FDA0002629228830000037
Monotonous decreasing region deltaMD
ΔMD＝max{ΔMD1，ΔMD2} (18)
Figure RE-FDA0002629228830000038
Wherein, DeltaMD1And ΔMD2Are all positive and real, and are determined by equation (19);
absolute attraction layer ΔAA
ΔAA＝max{ΔAA1，ΔAA2} (20)
Figure RE-FDA0002629228830000039
Wherein, DeltaAA1And ΔAA2Are all positive real numbers and are determined by equation (21);
steady state error band ΔSS
ΔSS＝max{ΔSS1，ΔSS2} (22)
Figure RE-FDA00026292288300000310
Wherein, DeltaSS1And ΔSS2Are all positive real numbers and are determined by equation (23);
maximum number of convergence steps
Figure RE-FDA0002629228830000041
Figure RE-FDA0002629228830000042
Wherein,
Figure RE-FDA0002629228830000043
represents the smallest integer no less than.
3. The method of claim 1 or 2, wherein the servo motor comprises a plurality of servo motors, each servo motor comprises: the adjustable parameter parameters of the controller comprise rho, alpha and mu; and the parameter setting of the controller is carried out according to the index representing the convergence process.
4. The method of claim 1 or 2, wherein the servo motor comprises a plurality of servo motors, each servo motor comprises: when the reference signal satisfies rk＝rk-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is dk＝wk-wk-1Wherein r isk-1Reference signal representing the time instant k-1, wk-1Representing the interference signal at time k-1; the feedback controller with equivalent disturbance compensation is
Figure FDA0002528166380000045
5. The design method of attraction law repetitive controller for specified time of servo motor as claimed in claim 1 or 2, characterized in that the adjustable parameters of the observer include β1And β2When k is sufficiently large, to ensure
Figure FDA0002528166380000046
And
Figure FDA0002528166380000047
respectively converge on ekAnd dkNeed to configure parameters β1，β2Make the matrix
Figure FDA0002528166380000048
Is within the unit circle.
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